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460 Chapter 6 Determinants

in its course, and major branches can dry up to become minor tributaries while
small trickling brooks can develop into raging torrents. This is precisely what
occurred with determinants and matrices. The study and use of determinants
eventually gave way to Cayley’s matrix algebra, and today matrix and linear
algebra are in the main stream of applied mathematics, while the role of deter-
minants has been relegated to a minor backwater position. Nevertheless, it is still
important to understand what a determinant is and to learn a few of its funda-
mental properties. Our goal is not to study determinants for their own sake, but
rather to explore those properties that are useful in the further development of
matrix theory and its applications. Accordingly, many secondary properties are
omitted or confined to the exercises, and the details in proofs will be kept to a
minimum.
Over the years there have evolved various “slick” ways to define the determi-
nant, but each of these “slick” approaches seems to require at least one “sticky”
theorem in order to make the theory sound. We are going to opt for expedience
over elegance and proceed with the classical treatment.
A permutation p =(p 1 ,p 2 ,...,p n ) of the numbers (1, 2,...,n) is simply
any rearrangement. For example, the set

{(1, 2, 3) (1, 3, 2) (2, 1, 3) (2, 3, 1) (3, 1, 2) (3, 2, 1)}

contains the six distinct permutations of (1, 2, 3). In general, the sequence
(1, 2,...,n) has n!= n(n − 1)(n − 2) ··· 1 different permutations. Given a per-
mutation, consider the problem of restoring it to natural order by a sequence
of pairwise interchanges. For example, (1, 4, 3, 2) can be restored to natural or-
der with a single interchange of 2 and 4 or, as indicated in Figure 6.1.1, three
adjacent interchanges can be used.
( 1, 4, 3, 2 )

( 1, 4, 3 2) ( 1, 4, 2, 3 )

( 1, 2, 3, 4 ) ( 1, 2, 4, 3 )

( 1, 2, 3, 4 )

Figure 6.1.1
The important thing here is that both 1 and 3 are odd. Try to restore
(1, 4, 3, 2) to natural order by using an even number of interchanges, and you
will discover that it is impossible. This is due to the following general rule that is
stated without proof. The parity of a permutation is unique—i.e., if a permuta-
tion p can be restored to natural order by an even (odd) number of interchanges,
then every other sequence of interchanges that restores p to natural order must